Analyzing the Convergence of the Adapted General Gauss-type Proximal Point Approach for Smooth Generalized Equations
- Md. Asraful Alom
- Md. Zaidur Rahman
- Md. Shaharul Islam
- Md. Modassir Adon
- 1298-1317
- May 24, 2025
- Engineering
Analyzing the Convergence of the Adapted General Gauss-type Proximal Point Approach for Smooth Generalized Equations
Md. Asraful Alom*, Md. Zaidur Rahman, Md. Shaharul Islam, Md. Modassir Adon
Department of Mathematics, Khulna University of Engineering & Technology, Bangladesh
*Corresponding author
DOI: https://doi.org/10.51244/IJRSI.2025.12040150
Received: 08 April 2025; Accepted: 21 April 2025; Published: 24 May 2025
ABSTRACT
This work analyzed the adapted general Gauss-type proximal point approach for solving smooth generalized equations like 
, where 
is a set-valued mapping with closed graph and 
is a single-valued mapping acting between two general Banach spaces 
and 
. To confirm the existence and convergence of any sequence produced by this technique under appropriate assumptions, we develop the convergence criteria of this approach by utilizing metrically regular mapping and gathered both semi-local as well as local convergence results. Lastly, we plot a numerical example to compare the semi-local convergence result of this technique.
Keywords: Metrically regular mapping, Set-valued mapping, Semi-local convergence, Fixed point lemma, Generalized Equation.
INTRODUCTION
The challenge of identifying a point 
also known as the solution of the problem satisfying the smooth Generalized Equation
(1)
is the focus of this thesis, where a single-valued function is smooth while stands for a set-valued map with closed graph, both and are Banach spaces and 
is an open subset of .
This dissertation deals with smooth generalized equations. Robinson first established the generalized equations as an absolute structure for a wide variety of variational issues in his works [1, 2], such as system of inequalities, system of nonlinear equations, complementary problems, equilibrium problems, variational inequalities, etc.; see in example [3-5]. Additionally, they have enough uses in applied computational sciences, economics, mathematical programming, traffic equilibrium problems, analysis of elastoplastic structures, etc.
In this research, we looked at two different convergent problems for iteratively solving generalized equations. The first of these is called semi-local convergence analysis and it focusses based on the data near the starting point 
, the convergence criterion is established. The second is called local convergence analysis and it concentrates on the convergence ball based on the data around a generalized equations solution. Among the most popular methods for solving the inclusion (1), the proximal point algorithm (PPA) is the one.
The PPA approach was first presented by Martinet [6] in 1970 for use in convex optimization. Rockafellar [7] examined the PPA under the general framework of monotone inclusion maximization. Numerous authors have investigated proximal point algorithm generalizations and discovered uses for this technique to solve particular variational issues in the past fifty-five years. The majority of the fast-expanding body of research on this topic has focused on various iterations of the algorithm for addressing monotone mapping-related issues, particularly monotone variational inequalities; see, for instance, [8,9]. Spingarn [10] was the first to study monotonicity in its weaker version; see Iusem et al. [11] for further information.
The generalized proximal point algorithm has been made available to Aragon Artacho et al. [12]. They have reported the local convergence analysis of the generic (PPA) for the mapping 
with set values under various metric regularity assumptions. For solving (1), Rashid et al. [13] gave the subsequent traditional Gauss-type proximal point algorithm (G-PPA). They obtained semi-local as well as local convergence findings in addressing problem (1). In his later work [5], Rashid created the G-PPA to solve variational inequality and produced results on both semi-local as well as local convergence.
Let 
be the neighbourhood of origin and select a sequence of continuous Lipschitz functions 
on with positive Lipschitz constants 
and 
Let 
stand for the subset of for any 
belongs to and for a certain sequence of scalars which are away from 0, that is described as bellow:
.
In order to solve the generalized smooth equation (1), Dontchev and Rockafellar developed the following PPA in [3]:
Algorithm 1 (PPA)
First step: Set 
, 
and 
Second step: Stop if 
; Otherwise, move on to Third step
Third step: Enter 
and if 
, select 
such
that 
.
Forth step: Compose 
.
Fifth step: Replace 
by 
prior to proceeding to Second step.
Keep in mind that not all of the sequences produced by Dontchev and Rockafellar [3] are convergent, and that they are not uniquely defined for a starting point close to a solution. Dontchev and Rockafellar [3] demonstrated that their algorithm produces a single, linearly convergent sequence to the solution under specific circumstances. Furthermore, it appears that the well-established approach of Alom and Rashid [14] is time-consuming. Some acceptable conditions can be utilized to avoid the sequences generated by the algorithm of Dontchev and Rockafellar [3] from all convergent. By using the proximal point strategy, they guarantee the validity of a single sequence that converges linearly to the outcome. In light of estimates using mathematics, this type of process is therefore unsuitable for mathematical applications. We look for an adapted general Gauss-type proximal point algorithm (GGPPA) in response to this obstacle. To solve the generalized equation (1) in the simplest manner, we suggest the adapted GGPPA with some novel concepts for the key theorem. We demonstrate this by substituting the metric regularity criterion for the Lipschitz-like feature.
Again, let 
denotes the subset of and is defined by
(2)
To show that every sequence generated by the adapted GGPPA exists and to show that it converges, Alom and Rashid [14] regarded as the primary thesis that 
and 
, during the fundamental theorem in this research, we believe that 
and 
. We show that our methodology for solving the smooth generalized equation (1) is better than the previous one. The difference between our proposed Approach 2 and Algorithm 1 is that the enhanced GGPPA generates sequences, all of which are convergent, whereas Algorithm 1 does not. Thus, the revised GGPPA that we recommend is provided below:
Algorithm 2 (Adapted GGPPA):
First step: Set , and 
and 
Second step: Stop if 
; Otherwise, move on to Third step.
Third step: Enter , 
and if 
, select 
such
that 
and 
.
Forth step: Compose 
.
Fifth step: Replace by prior to proceeding to Second step.
Based on Algorithm 2, we note that
- When
and 
is singleton, Algorithm 2 becomes identical to Algorithm 1. - According to Alom et al. [15], the generalized Gauss-type proximal point technique is similar to Algorithm 2 if
, - if
, Algorithm 2 is equivalent to the GPPA for solving smooth generalized equation introduced by Alom and Rashid [16].
For solving (1) in the situation and analyzing the results of semi-local and local convergence, Alom et al. [15] developed the generic version of the G-PPA. Alom et al. [17] established the uniformity of the GG-PPA of (1) with situation 
. In order to solve the variational inequality problem, Rashid [5] developed the Gauss-type proximal point approach, which produced the result of semi-local as well as local convergence. Alom et al. [18] recently proposed a adapted general form of the Gauss-type proximal point algorithm (GG-PPA) to address the generalized equations in the case where 
They also conducted an analysis of the algorithm’s local and semi-local convergence properties. Also, Alom and Rahman [19] introduced the stability analysis of adapted GG-PPA for solving (1) in the case using metrically regular mapping. Alom and Rashid [16] introduced the GPPA for the purpose of solving the generalized smooth equation (1) with the combinations of metrically regular mapping and Lipschitz-like continuity. Next time, Alom and Rashid [14] introduced the GGPPA in order to solve the generalized smooth equation (1) with the combinations of metrically regular mapping and Lipschitz-like continuity and obtained both the result of semi-local as well as local convergence. Alom et al. [20] introduced the adapted GPPA for solving the generalized smooth equation (1) and obtained the result of semi-local as well as local convergence.
To the best of our knowledge, no research has been done on semi-local analysis to solve the above generalized equations by using only metrically regular mappings which motivates us to research in this field to extend the idea. We suggest the adapted GG-PPA for resolving (1) with some changes to the vital theorem of [14] and verify this by applying metric regularity condition instead of Lipschitz-like property. We show that our approach outperforms the previous method in solving (1)
Notations and Preliminaries
Several conventional notations, basic ideas, and mathematical conclusions that will frequently be cited in the
next section is reviewed in this section of the article. It is assumed that both and are general Banach spaces. The formula declare the set valued mapping from to a subset of . Allow 
and 
a closed ball with radius 
and centre at is indicated by the symbol 
.
The graph of terms is expressed by and symbolized by 
.
,
the domain of is represented by 
and is expressed by
,
and 
is the inverse of , which can be stated as
.
By 
, the symbol stands all the norms. Permit 
and 
to be, respectively, subsets of .
in all cases of 
,
specifies the separation between and , where as
the excess between sets and is defined.
Definition 1. Inner product space: If there is a complex number 
associated with every pair of vectors 
so that the bellow properties are true, then a complex vector space as an inner product space, 
is moved to:
, the complex-conjugate is shown by the bar,
,
and 
iff 
.
Definition 2. Hilbert space: If a complex inner product space is complete with regard to the metric that its inner product induces, it is referred to be a Hilbert space. The metric is produced by the inner product norm.
Definition 3. Normed Linear space: A space that is linear if any vector 
in has a real number associated with it, denoted by the symbol “
” (also known as the norm of ), then is said to be a normed linear space.
, for any 
and 
,
, and 
iff 
,
for any 
.
Definition 4. Banach space: If under the metric generated by the norm, a normed linear space is complete as a metric space, it is referred to as a Banach space.
Parallelogram law: In a Hilbert space, if and 
are two vectors, then
.
Remark 1. The Banach space 
converts to a Hilbert space if the norm of it complies with the parallelogram law and the inner product on B is expressed by
.
We agree with the definition of mathematically regularity from [5] for mapping with set values.
Definition 5. Metrically regular Mapping:
Assuming 
, where 
denotes a set-valued mapping. Let 
, 
, and 
all be greater than zero. When
for each 
, 
, (3)
then at 
on 
relative to 
with constant 
one can say that the mapping 
is mathematically regular.
We revisit the ideas of Lipchitz-like continuity from [13] for set-valued mappings. Aubin first proposed this concept in [21].
Definition 6. Lipschitz-like continuity:
Let (
) be a member of gphγ and 
be a mapping with set-values. Assume that 
, 
and 
are all greater than 
. If the following discrepancy occurs, the mapping 
is considered Lipschitz-like for any 
belongs to 
, then at 
on 
relative to 
together with constant ,
(4)
We obtain the following lemma from [15], which proves the connection between a mapping of metric regularity at 
) and the Lipschitz-like continuity of the inverse at .
Lemma 1. Let 
where be a function with set values. Take and both are not less than zero. It becomes at 
on 
relative to 
the function is metrically regular with constant 
for any 
, iff at 
on relative to the inverse 
is Lipschitz-like with constant , that is,
(5)
Proof: Consider that at on relative to metrically the mapping Q is regular and is constant. Consider belongs to . We must demonstrate that (5) is true. To be able to demonstrate this, suppose belongs to 
. We find
(6)
because at on relative to metrically the mapping Q is regular and is constant. Therefore
according to the definition of access e. This results in the statement that
,
coupled with (6). This suggests that (5) is met.
Consider, however, that (5) is true. We must demonstrate that at on relative to 
the mapping 
is metrically regular with being constant. Let 
belongs to and 
belongs to
to finish this. Given that (5) is valid for 
belongs to, let belongs to 
. As a result, 
belongs to . The result
is then obtained from the definition of excess 
. From the inequality above, if we select the minimum with regard to belongs to on both sides, we obtain
which is true for any values of belongs to and belongs to . As a result, it can be seen that at on relative to , is constant and the mapping Q is metrically regular. As a result, the proof of Lemma-1 is finished.
The Lyusternik-Graves theorem was borrowed from [22]. We carry out that a set if 
is bigger than , then 
belongs to 
, which is a locally closed subset of . As a result, the set 
is closed.
Lemma 2. Lyusternik-Graves theorem: Assuming that 
, 
consideration is given to 
being locally closed and being a mapping with set values. For any 
let be metrically regular at 
and have constant 
. Consider a function 
that is continuous at and with a Lipschitz constant 
such that is less than 
. For 
, the mapping 
is metrically regular at
and has constant 
.
In [23], Dontchev and Hagger established the fixed-point lemma for set-valued mappings, which generalized the fixed-point theorem from [16]. This lemma is indispensable for proving the existence of any sequence.
Lemma 3. Banach fixed point Lemma:
Assume that 
is a mapping with predetermined values. Assume that belongs to 
, 
belongs to ,, and 
is such that
(7)
and for all 
,
(8)
are each satisfied. This means that 
has a fixed point in 
, indicating that belongs to exists and 
. There is just one fixed point of 
in , if is also single-valued.
Convergence Analysis of the adapted GGPPA
Consider that both X and Y are general Banach spaces. Let 
be a mapping with set values which has locally closed graph, and let 
be a smooth function on 
. Let , , 
, and all be greater than 0 such that 
is less than 1. We define
(9)
It is evident from (9) that 
and 
.
To prove the adapted GGPPA’s semi-local convergence conclusion, we employ the following lemma.
Lemma 4. The set valued mapping should have a locally closed graph at 
. To define 
, use (9). Assume that with constant the mapping 
is metrically regular at on 
. With 
, let be the mapping with Lipschitz continuous on 
with Lipschitz constant. Then at on 
with constant 
, the function 
is metrically regular.
Proof. We find that
for any 
, 
based on our assumption regarding 
. We will demonstrate that
for any 
and 
For the purpose of completing this, we’ll start with the induction of and confirm that a sequence 
, with 
, like that, for 
; meets the subsequent claims:
(10)
and
(11)
It is evident that (10) holds true when . Using the second condition in clause (9), we have 
and for 
, 
is positive. This implies that 
that is, 
Hence, for any belongs to 
and 
belongs to 
we observe that
+
(12)
It becomes 
belongs to 
. We have 
as has a locally closed graph with 
This demonstrates that (10) is real for . Additionally, we are able to write
(13)
by utilizing the ‘s metric regularity condition. Moreover,
(14)
Therefore
(15)
As , we obtain that
based on the first condition in (9). Thus, we write from (15) that
This becomes 
. By applying (15), we are able to type that
+
(16) We discover from the second requirement in clause (9) that
and as 
suggests a positive number less than 1 , it follows that
, that is
So, we get from (16) that
It becomes 
is true. Given that has a locally closed graph, it is evident that 
and (10) is accurate for equal to 1. We can now write by using 
equal to 
and the metric regularity assumption on that
(17)
From (14) and (17), We are able to write
)
( 
) (18)
Using 
and 
for any values of 
such that <1 and the first condition in (9), we gain from (18) that
This becomes 
. We are able to write by applying the metric regularity condition on that
. So, for 
, (11) is accurate. This demonstrates that (10), (11) holds for the built-in points 
when 
. Suppose 
are built so that (10) and (11) are applicable to 
We must make 
sure that (10) and (11) are valid for 
by induction hypothesis. First, we’ll demonstrate that 
belongs to 
for all 
Inferring that from (11)
(19)
In addition, we are able to write
=
(20)
by using (19), (14), and the first condition in (9). From (20) for 
equal to 
and by applying the second circumstance in (9), it demonstrates that belongs to 
for any 
and
Therefore, 
belongs to 
. As a result, 
because the graph of is locally closed. It suggests that (10) holds true when . Applying the metric regularity assumption on , we arrive at
(21)
Due to the completion of the induction steps, (10) and (11) are true for all . We discover from (21) that 
(22)
Applying the relation 
from (20), we can determine from (22) that
+
Consequently, belongs to . Thus {
is a sequence of Cauchy and all of its members are in , as we can see from (21). Then, assuming the limit in (10) and the local closeness of satisfying 
i.e., 
i.e., the sequence ends up at some 
, i.e., 
Using (11) and (13), we discover
As a result, the Lemma 4’s proof is finished.
Let’s say that 
is metrically regular at 
on 
with constant and 
is closed. Consider a single valued function 
with 
, it has a Lipschitz constant and is Lipschitz continuous about the origin, meaning that 
. Define a mapping 
by
for any .
Then for any 
and 
we obtain
(23)
In particular, 
for each 
(24)
Here 
is Lipschitz continuous on 
with constant 
Lemma 4 is therefore applied, and we assume that the mapping 
is metrically regular at 
on with constant 
. So, by Lemma 1, we say that the mapping 
is Lipschitz-like at 
on 
with constant 
that is,
for all 
(25)
Suppose that
(26)
Write
. (27)
Then
. (28)
To prove the convergence result of the adapted GGPPA, we need the following lemma. The refinement of the evidence for [35] serves as the proof.
Lemma 5. Given a constant 
, let 
be metrically regular at on such that (27) and (28) are fulfilled. Consider 
and 
. Then, 
is Lipschitz-like at on 
with constant 
, that is, 
for all 
In order to finish our primary conclusion, assuming a series of functions such that are Lipschitz constants are fulfilled by Lipschitz continuity near the origin, which is identical for all .
(29)
When we swap out 
in (23) for 
, we get the mapping 
as follows:
for each 
(30)
and rewrite equation (25) in the manner shown below:
for all (31)
Then, we have from (2) that
(32)
Again, we specify the mapping 
by
for each 
(33)
Additionally, the set valued mapping 
by
(34)
Thus, for each 
||
(35)
We now give the following essential result and its proof given a set of suitable conditions with initial point 
, arbitrary sequence produced by the GGPPA is guaranteed to exist and to converge semi-locally.
Theorem 1. Let’s say that is metrically regular at on with constant and 
is closed. Consider a single valued function with , which holds the Lipschitz continuity property around the origin with Lipschitz constant such that . Allow 
to be such that
-
, - ,
- .
Then there is some 
is greater than zero so that one or more sequences 
are produced using Algorithm 2 and every sequence that is produced converges to a solution 
of (1), i.e., 
verifies that 
.
Proof. Consider that
Then by applying the assumption 
with 
, we have
Assumptions 
and 
allow us to take 
such that for each 
. (36)
By mathematical induction, we will then show that Algorithm 2 generates many sequences, with each sequence generated by Algorithm 2 satisfies the followings:
(37)
and
for every 
(38)
In order to prove the inequalities (37) as well as (38), we define the constant 
by
(39)
By the assumptions and with , we have
for every 
). (40)
Clearly (37) holds for . To prove that (38) is valid for , we must establish that 
exists i.e., 
. To do this, we will consider the mapping 
defined by (34) and apply Lemma 3 to with 
and 
. It is sufficient to show that axioms (7) and (8) of Lemma 3 are valid for with and . By the definition of in (34), we are able to write 
. Therefore, we obtain
. (41)
Now that we know what metric regularity is, we are able to write
(42)
as 
according to the set valued mapping definition 
Consequently, we derive (41) and (42)
(43)
Using the selection of λ and the concept of Lipschitz continuous mapping,
we derive from the mapping’s definition 
in (33) that
||
||
||. (44)
As 
, then by the assumption 
in (a) and by the assumption 
|
in (c), we write from (44) that
. (45)
This shows that for every 
, 
More specifically,
||
||
|| (46)
.
This implies that 
. By using (42) in (43), we obtain
(47)
By using (47) in (39) with and 
, we get
This demonstrates that axiom (7) of Lemma 3 is valid. Now, we demonstrate that axiom (8) of Lemma 3 is also valid. For this, consider 
belongs to 
. Therefore, we get 
by using (40). By the first assumption 
in (a), We are able to write 
. Then from (45), we obtained that 
. By using the characterization of the set-valued mapping from (34) and using the concept of metric regularity, we can express the relationship as
(48)
Now, by using (35) in (48) and by the definition of Lipschitz continuous function, we observe that
. (49)
By assumption (b), the above inequality becomes
Thus, the axiom (8) of Lemma 3 is also valid. Given that axioms (7) and (8) of Lemma 3 are valid, We can determine that a fixed point exists 
so that 
, that translates to 
that is 
. This shows that 
and thus . Consequently, as 
we can choose 
such that
. (50)
By Algorithm 2, 
is specified. Therefore, the point 
is generated by Algorithm 2. Additionally, from the definition of 
, from (2) we are able to write
and so
Since 
is metrically regular at on relative to with constant 
as a consequence of Lemma 5 the mapping 
is Lipschitz-like at on 
relative to 
possessing the constant for every 
.More specifically, 
is Lipschitz-like at on 
relative to with constant as the ball 
contains the point 
Furthermore, by the assumption in (c) and the assumption 
in (a), we obtain that
It shows that 
According to Lemma 5, with constant 
, the mapping 
is metrically regular at 
) on relative to 
Therefore, using Lemma 1, we get
(51)
The equation (36) implies that
(52)
As a result, we conclude that
, (53)
which we get from (32) and use in (52). By using (53), we obtain from Algorithm 2 that
(54)
It follows from this that (38) is valid for .
Assuming that Algorithm 2 produced the coordinates 
we can conclude that (37), (38) hold for 
. We demonstrate that there exists such that (37), (38) are satisfied for . The assumptions (37) and (38) are real for every 
, Consequently, we derive the relationship
(55)
and so 
It shows that (37) is valid for . Using much the same reasoning as when 
We can deduce that the mapping 
is Lipschitz-like at on relative to with constant . Then, by using algorithm 2 once more, we have
(56)
.
This demonstrates that (38) is real for . Therefore (37), (38) are valid for every . This suggests that is a Cauchy sequence which is produced by Algorithm 2 and hence there exists 
such that 
Thus, passing to the limit 
and since 
is closed, it follows that 
Thus the proof is completed.
Imagine that 
(57)
Theorem 1 is simplified to the subsequent consequence, it explains how the Algorithm 2 sequence locally converges, when is a special case solution of (1), that is,
.
Corollary 1 Let’s say that 
is satisfied and that , 
. Assume that has a locally closed graph with constant 
at 
and is metrically regular there. Select a scalar sequence of length . Then, there exists 
such that every sequence produced by Algorithm 2 with the beginning point 
terminates to a solution satisfying that .
Proof. According to our presumption, is metrically regular at , where a locally closed graph with constant exists. Then, there are constants 
and 
such that with constant , is metrically regular at on 
) (0), indicating that the inequality below is valid.
for all 
.
Think of 
as being such that and 
. For every 
close to origin so that 
is locally closed at 
, since is very close to 
Allow us to take in order to achieve
for any 
. Since is metrically regular at on 
with constant , one gets that
for all 
where 
such that
and 
Then
and
We can therefore select 
so that
.
It is now common practice to check that Theorem 1’s assumptions are all true. Therefore, we may finish the proof of the corollary by using Theorem 1
Numerical Test
A numerical test is provided in this part to validate the result of semi-local convergence of the adapted GGPPA
Example 1 Consider 
, 
, 
and 
Select a set-valued mapping on 
by 
and a smooth function 
on by 
. Also, choose a Lipchitz continuous function by 
=
, where The mapping with set-values is thus defined as 
on . Afterward, Algorithm 2 yields a sequence that eventually meets to 
.
Take into consideration 
. Then from the statement, it is clear that is metrically regular at 
and is Lipschitz continuous in the neighbourhood of origin with Lipschitz constant 
. Then from (1), we have that
.
However, if 
, we get that
According to this,
.
As a result, we deduce from (56) that
We can observe that for the specified values of 
and 
. This demonstrates that the sequence created by Algorithm 2 meets linearly, supporting the algorithm’s result of semi-local convergence. The generalized equation has a solution 0.25 for 
according to the following table 1, which was generated by the Matlab application.
|
|
|
0.2000 0.2000 0.2545 -0.0182 0.2496 0.0017 0.2500 -0.0002 0.2500 0.0000 |
Table 1: Identifying a solution
The graphic depiction of 
is shown in the following figure:
Figure 1: The graph of .
Final Observations
The semi-local as well as local convergence findings for the adapted GGPPA specified by Algorithm 2 have been developed in this work. The generalized smooth equation (1) can be solved with the following assumptions: is a smooth differentiable function, is a set valued mapping that is metrically regular, has a locally closed graph, and is a sequence of Lipschitz continuous functions such that around the origin with Lipschitz constant 
In the event when 
, and 
is a singleton, the findings of this study align with those found in [3]. We have supported the study of semi-local convergence of the adapted GGPPA with a numerical example. The outcome builds upon and enhances the outcome found in [3, 14]. Next time, we will try to analyze the convergence of the Gauss-type proximal point method for non-smooth generalized equations.
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